#### Abstract

We have investigated the periodical solutions of the system of rational difference equations , and where .

#### 1. Introduction

Recently, a great interest has arisen on studying difference equation systems. One of the reasons for that is the necessity for some techniques which can be used in investigating equations which originate in mathematical models to describe real-life situations such as population biology, economics, probability theory, genetics, and psychology. There are many papers related to the difference equations system.

In [1], Kurbanli et al. studied the periodicity of solutions of the system of rational difference equations

In [2], Çinar studied the solutions of the systems of difference equations

In [3, 4], Özban studied the positive solutions of the system of rational difference equations

In [5–16], Elsayed studied a variety of systems of rational difference equations; for more, see references.

In this paper, we have investigated the periodical solutions of the system of difference equations where the initial conditions are arbitrary real numbers.

#### 2. Main Results

Theorem 1. *Let , , , , , , , , and be arbitrary real numbers, and let be a solution of the system
**
Also, assume that , , , and . Then, all six-period solutions of (5) are as follows:
*

*Proof. *For , we have
For , assume that
are true. Also, we have

Theorem 2. *Let , , , , , , , , and be arbitrary real numbers, and let be a solution of the system
**
Also, assume that , , , and . Then, all six-period solutions of (10) are as follows:
*

*Proof. *For , we have
For , assume that
are true. Also, we have

The following corollary follows from Theorem 1.

Corollary 3. *The following conclusions are valid for :*(i)*,*(ii)*,*(iii)*,*(iv)*.*

The following corollary follows from Theorem 2.

Corollary 4. *The following conclusions are valid for :*(i)*,*(ii)*,*(iii)*,*(iv)*.*